The ability to organize and finely manipulate the hierarchy and timing of dynamic processes is important for understanding and influencing brain functions, sleep and metabolic cycles, and many other natural phenomena. However, establishing spatiotemporal structures in biological oscillator ensembles is a challenging task that requires controlling large collections of complex nonlinear dynamical units. A method is presented to design entrainment signals that create stable phase patterns in ensembles of heterogeneous nonlinear oscillators without using state feedback information. This approach is demonstrated using experiments with electrochemical reactions on multi-electrode arrays, in which ensemble subgroups are selectively assigned into spatiotemporal patterns with multiple phase clusters. The experimentally confirmed mechanism elucidates the connection between the phases and natural frequencies of a collection of dynamical elements, the spatial and temporal information that is encoded within this ensemble, and how external signals can be used to retrieve this information.
Complex interactions among nonlinear periodic phenomena emerge in many natural and engineered systems. Numerous instances appear in chemical reactions and biological systems, which exhibit endogenous and emergent multi-scale oscillation. There is significant interest in characterizing synchronization in oscillators interconnected in networks, which is especially important for understanding the highly complex dynamics of man-made systems such as electric power grid, and elucidating the functions of neural systems. Understanding entrainment of oscillating systems to an exogenous forcing signal is crucial to modeling circadian timekeeping, dynamic neural regulation, and for the design of synchronizing or desynchronizing treatments of cardiac arrhythmias, Parkinson's disease, epilepsy, and movement disorders.
Even the simplest models of interacting oscillators can exhibit highly complex behavior, and individual oscillating units may themselves possess complicated dynamics. These factors are aggravated in practice by model and parameter uncertainty and the impracticality of obtaining feedback information, such as for in vivo biological applications, and pose challenges to manipulating or controlling oscillating ensembles. Such tasks require tractable yet accurate simplifications of the complex dynamic interactions involved, and demand suitable mathematical approaches that characterize ensemble-level properties while withstanding experimental uncertainties.
Control-theoretic techniques have been applied to control a single oscillator. In contrast, finely manipulating individual sub-systems in under-actuated ensembles, such as thousands of neurons in the brain affected by one electrode, rather than activating them homogeneously, remains a fundamental challenge. Synchronization has been engineered in collections of oscillators using feedback, or tuning coupling strengths. Such approaches require certain coupling structures, exact model specification, state feedback information, or precise knowledge of initial conditions, but still are not able to produce a prescribed phase pattern corresponding to frequency clusters of the oscillators.
Versatile open-loop control techniques were developed for simultaneous control of ensembles of quantum spin systems, which motivated the field of ensemble control. Inspired by selective pulse design in nuclear magnetic resonance (NMR), which enabled revolutionary applications including functional magnetic resonance imaging (fMRI), a method is developed for selectively manipulating the subunits of oscillator ensembles using periodic inputs that are robust to parameter uncertainty and disturbances. Specifically, the method exploits the slight heterogeneity and high nonlinearity of an ensemble of structurally similar oscillators far past the Hopf bifurcation, rather than relying on a known coupling structure, state feedback, or initial condition information.